Best Known (40, 40+17, s)-Nets in Base 3
(40, 40+17, 144)-Net over F3 — Constructive and digital
Digital (40, 57, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 19, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
(40, 40+17, 182)-Net over F3 — Digital
Digital (40, 57, 182)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(357, 182, F3, 17) (dual of [182, 125, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(357, 253, F3, 17) (dual of [253, 196, 18]-code), using
- construction XX applied to C1 = C([106,121]), C2 = C([108,122]), C3 = C1 + C2 = C([108,121]), and C∩ = C1 ∩ C2 = C([106,122]) [i] based on
- linear OA(351, 242, F3, 16) (dual of [242, 191, 17]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {106,107,…,121}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(351, 242, F3, 15) (dual of [242, 191, 16]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {108,109,…,122}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {106,107,…,122}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(346, 242, F3, 14) (dual of [242, 196, 15]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {108,109,…,121}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(31, 6, F3, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(30, 5, F3, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([106,121]), C2 = C([108,122]), C3 = C1 + C2 = C([108,121]), and C∩ = C1 ∩ C2 = C([106,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(357, 253, F3, 17) (dual of [253, 196, 18]-code), using
(40, 40+17, 4108)-Net in Base 3 — Upper bound on s
There is no (40, 57, 4109)-net in base 3, because
- 1 times m-reduction [i] would yield (40, 56, 4109)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 523 527573 322016 311826 747745 > 356 [i]